Optimal. Leaf size=179 \[ \frac {6 (c x)^{-n (p+4)} \left (a+b x^n\right )^{p+4}}{a^4 c n (p+1) (p+2) (p+3) (p+4)}-\frac {6 (c x)^{-n (p+4)} \left (a+b x^n\right )^{p+3}}{a^3 c n (p+1) (p+2) (p+3)}+\frac {3 (c x)^{-n (p+4)} \left (a+b x^n\right )^{p+2}}{a^2 c n (p+1) (p+2)}-\frac {(c x)^{-n (p+4)} \left (a+b x^n\right )^{p+1}}{a c n (p+1)} \]
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Rubi [A] time = 0.13, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {273, 264} \[ \frac {3 (c x)^{-n (p+4)} \left (a+b x^n\right )^{p+2}}{a^2 c n (p+1) (p+2)}-\frac {6 (c x)^{-n (p+4)} \left (a+b x^n\right )^{p+3}}{a^3 c n (p+1) (p+2) (p+3)}+\frac {6 (c x)^{-n (p+4)} \left (a+b x^n\right )^{p+4}}{a^4 c n (p+1) (p+2) (p+3) (p+4)}-\frac {(c x)^{-n (p+4)} \left (a+b x^n\right )^{p+1}}{a c n (p+1)} \]
Antiderivative was successfully verified.
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Rule 264
Rule 273
Rubi steps
\begin {align*} \int (c x)^{-1-4 n-n p} \left (a+b x^n\right )^p \, dx &=-\frac {(c x)^{-n (4+p)} \left (a+b x^n\right )^{1+p}}{a c n (1+p)}-\frac {3 \int (c x)^{-1-4 n-n p} \left (a+b x^n\right )^{1+p} \, dx}{a (1+p)}\\ &=-\frac {(c x)^{-n (4+p)} \left (a+b x^n\right )^{1+p}}{a c n (1+p)}+\frac {3 (c x)^{-n (4+p)} \left (a+b x^n\right )^{2+p}}{a^2 c n (1+p) (2+p)}+\frac {6 \int (c x)^{-1-4 n-n p} \left (a+b x^n\right )^{2+p} \, dx}{a^2 (1+p) (2+p)}\\ &=-\frac {(c x)^{-n (4+p)} \left (a+b x^n\right )^{1+p}}{a c n (1+p)}+\frac {3 (c x)^{-n (4+p)} \left (a+b x^n\right )^{2+p}}{a^2 c n (1+p) (2+p)}-\frac {6 (c x)^{-n (4+p)} \left (a+b x^n\right )^{3+p}}{a^3 c n (1+p) (2+p) (3+p)}-\frac {6 \int (c x)^{-1-4 n-n p} \left (a+b x^n\right )^{3+p} \, dx}{a^3 (1+p) (2+p) (3+p)}\\ &=-\frac {(c x)^{-n (4+p)} \left (a+b x^n\right )^{1+p}}{a c n (1+p)}+\frac {3 (c x)^{-n (4+p)} \left (a+b x^n\right )^{2+p}}{a^2 c n (1+p) (2+p)}-\frac {6 (c x)^{-n (4+p)} \left (a+b x^n\right )^{3+p}}{a^3 c n (1+p) (2+p) (3+p)}+\frac {6 (c x)^{-n (4+p)} \left (a+b x^n\right )^{4+p}}{a^4 c n (1+p) (2+p) (3+p) (4+p)}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 69, normalized size = 0.39 \[ -\frac {x (c x)^{-n (p+4)-1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (-p-4,-p;-p-3;-\frac {b x^n}{a}\right )}{n (p+4)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 294, normalized size = 1.64 \[ -\frac {{\left (6 \, a b^{3} p x x^{3 \, n} e^{\left (-{\left (n p + 4 \, n + 1\right )} \log \relax (c) - {\left (n p + 4 \, n + 1\right )} \log \relax (x)\right )} - 6 \, b^{4} x x^{4 \, n} e^{\left (-{\left (n p + 4 \, n + 1\right )} \log \relax (c) - {\left (n p + 4 \, n + 1\right )} \log \relax (x)\right )} - 3 \, {\left (a^{2} b^{2} p^{2} + a^{2} b^{2} p\right )} x x^{2 \, n} e^{\left (-{\left (n p + 4 \, n + 1\right )} \log \relax (c) - {\left (n p + 4 \, n + 1\right )} \log \relax (x)\right )} + {\left (a^{3} b p^{3} + 3 \, a^{3} b p^{2} + 2 \, a^{3} b p\right )} x x^{n} e^{\left (-{\left (n p + 4 \, n + 1\right )} \log \relax (c) - {\left (n p + 4 \, n + 1\right )} \log \relax (x)\right )} + {\left (a^{4} p^{3} + 6 \, a^{4} p^{2} + 11 \, a^{4} p + 6 \, a^{4}\right )} x e^{\left (-{\left (n p + 4 \, n + 1\right )} \log \relax (c) - {\left (n p + 4 \, n + 1\right )} \log \relax (x)\right )}\right )} {\left (b x^{n} + a\right )}^{p}}{a^{4} n p^{4} + 10 \, a^{4} n p^{3} + 35 \, a^{4} n p^{2} + 50 \, a^{4} n p + 24 \, a^{4} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{-n p - 4 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \left (c x \right )^{-n p -4 n -1} \left (b \,x^{n}+a \right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{-n p - 4 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,x^n\right )}^p}{{\left (c\,x\right )}^{4\,n+n\,p+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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